(C) The mass $m$ moves in a horizontal circle of radius $r = \ell \sin \theta$. The angular frequency is $\omega = 2\pi f = 2\pi \times \frac{1}{\pi}

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(C) The mass $m$ moves in a horizontal circle of radius $r = \ell \sin \theta$. The angular frequency is $\omega = 2\pi f = 2\pi \times \frac{1}{\pi} = 2 \text{ rad/s}$. The forces acting on the mass are tension $T$ (along the string) and weight $mg$ (downwards). Resolving tension into components: $T \cos \theta = mg$ and $T \sin \theta = m\omega^2 r$. Substituting $r = \ell \sin \theta$,we get $T \sin \theta = m\omega^2 \ell \sin \theta$,which simplifies to $T = m\omega^2 \ell$. Substituting $\omega = 2 \text{ rad/s}$,we get $T = m(2)^2 \ell = 4m\ell$.

Class 11 Physics Newton's Laws of Motion and Friction Circular motion with Friction

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An inextensible string of length $\ell$ fixed at one end,carries a mass $m$ at the other end. If the string makes $\frac{1}{\pi}$ revolutions per second around the vertical axis through the fixed end,the tension in the string is [The string makes an angle $\theta$ with the vertical]

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A
$4\pi^2 m\ell$
B
$2\pi^2 m\ell$
C
$4 m\ell$
D
$m\ell$

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Newton's Laws of Motion and Friction

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Circular motion with Friction

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An inextensible string of length $\ell$ fixed at one end,carries a mass $m$ at the other end. If the string makes $\frac{1}{\pi}$ revolutions per second around the vertical axis through the fixed end,the tension in the string is [The string makes an angle $\theta$ with the vertical]

Similar Questions

If the coefficient of friction between the tires and the road is $\mu$,the maximum safe speed is $10\;m/s$. If the coefficient of friction becomes $\mu' = \frac{\mu}{2}$,what will be the new maximum safe speed?

$A$ car is moving along a circular path having a coefficient of friction $0.5$ and a radius of curvature $16.2 \,m$. The maximum velocity of the car that can travel without skidding outwards is (Acceleration due to gravity $= 10 \,ms^{-2}$)

$A$ block $(P)$ is rotating in contact with the vertical wall of a rotor as shown in figures $A$,$B$,and $C$. Find the relation between angular velocities $\omega_A, \omega_B$,and $\omega_C$ such that the block does not slide down. ($R_A < R_B < R_C$ are the radii).

$A$ car is moving on a horizontal curved road with radius $50\,m$. The approximate maximum speed of the car will be $............\,ms^{-1}$,if the coefficient of friction between the tyres and the road is $0.34$. [Take $g = 10\,ms^{-2}$]

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